Motives on general base "spaces"
Friday, 22.3.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk I will begin with an introduction to motives. I will then\ndefine a category of motives on very general base spaces, so-called\nprestacks. This framework allows us to consider motives on, say, an\ninfinite-dimensional affine space, and also equivariant motives. If time\npermits I will sketch an application of this formalism to a motivic\nSatake equivalence, a cornerstone in the Langlands program. My intention\nis to keep the talk as non-technical as possible. This is joint work\nwith Timo Richarz.
"Being faster by disrespecting the elder rule!" --- Why Discrete Morse Theory improves Persistent Homology computation
Thursday, 28.3.19, 10:15-11:15, Raum 404, Ernst-Zermelo-Str. 1
Persistent homology is a tool for topological data analysis, that can help to\nanalyse deformed geometric shapes like connected components, circles, voids and\nhigher dimensional homology. The computation of persistent homology is based on\nthe construction of a filtered cell complex and scales roughly cubic in the\nnumber of cells. Discrete Morse theory reduces the number of cells in a complex\nwithout changing its homology. In 2013 Vidit Nanda and Konstantin Mischaikow\nused filtration-wise Morse reductions to proof a speed up for certain\npersistent homology computations and implemented the software Perseus.\n\nIn practice, many filtered cell complexes grow by one simplex per filtration\nvalue and cannot be reduced by Nanda and Mischaikow's approach, e.g. Cech\ncomplexes. This talk will show some ideas how to trade off an approximated\nresult for a faster computation. This effect can be explained by allowing small\ndeviations from the elder rule. The new construction of an induced filtered\nacyclic matching helps for an informed choice of the approximation parameter.\nAlso, the theoretical construct of pairings on a graded multiset of real\nnumbers unifies persistent homology and filtered acyclic matchings. As an aside\nthis allows the purely combinatorial proof of a filtered Euler formula for all\nsuch pairings.\n